Is there a way to stop Chrome/Firefox record history of a 'particular' website?

I want Chrome/Firefox NOT to record history of a 'particular' website. Is it possible to do so? I searched in preferences, but couldn't find a solution.

This isn't the exact solution you want, but I couldn't find another way to do it more in line with the original question.

Chrome

The solution is to use incognito mode. To get to incognito mode, use the following keyboard shortcut:

Ctrl+Shift+N

Firefox

The solution is to use private browsing. To start private browsing, use the following keyboard shortcut:

Ctrl+Shift+P

In case of chrome there is an Extension called "Ghost Incognito", Ghost incognito make sure that some domains always open in incognito mode. When navigating to a listed domain, the tab will open in a new incognito window.

You could also use these

history site blocker (for Chrome)

Note that while sites are prevented from being inserted into the history, autocomplete still works on URLs from those sites. I actually like the convenience of this, but it's definitely less secure than Incognito, which prevents those URLs from showing up in autocomplete.

History blocker (for Firefox)